Hierarchical identity-based encryption and signature schemes

ABSTRACT

Methods are provided for encoding and decoding a digital message between a sender and a recipient in a system including a plurality of private key generators (“PKGs”). The PKGs include at least a root PKG and n lower-level PKG in the hierarchy between the root PKG and the recipient. A root key generation secret is selected and is known only to the root PKG. A root key generation parameter is generated based on the root key generation secret. A lower-level key generation secret is selected for each of the n lower-level PKGs, wherein each lower-level key generation secret is known only to its associated lower-level PKG. A lower-level key generation parameter also is generated for each of the n lower-level PKGs using at least the lower-level key generation secret for its associated lower-level private key generator. The message is encoded to form a ciphertext using at least the root key generation parameter and recipient identity information associated with the recipient. A recipient private key is generated such that the recipient private key is related to at least the root key generation secret, one or more of the n lower-level key generation secrets, and the recipient identity information. The ciphertext is decoded to recover the message using at least the recipient private key.

RELATED APPLICATIONS

The present application is a division of U.S. patent application Ser. No. 10/384,328, filed May 7, 2003, incorporated herein by reference, which claims priority under 35 U.S.C. §119(e) to provisional U.S. patent applications Ser. No. 60/366,292, filed on Mar. 21, 2002, and Ser. No. 60/366,196, filed on Mar. 21, 2002, both of which are incorporated herein by reference.

BACKGROUND OF THE INVENTION

The present invention relates in general to cryptography and secure communication via computer networks or via other types of systems and devices, and more particularly to hierarchical, identity-based schemes for encrypting and decrypting communications.

Roughly speaking, identity-based cryptosystems are public key cryptosystems in which the public key of an entity is derived from information associated with the entity's identity. For instance, the identity information may be personal information (i.e., name, address, email address, etc.), or computer information (i.e., IP address, etc.). However, identity information may include not only information that is strictly related to an entity's identity, but also widely available information such as the time or date. That is, the importance of the concept of identity information is not its strict relation to the entity's identity, but that the information is readily available to anyone who wishes to encrypt a message to the entity.

An entity's private key is generated and distributed by a trusted party or logical process, typically known as a private key generator (“PKG”). The PKG uses a master secret to generate private keys. Because an entity's public key may be derived from its identity, when Alice wants to send a message to Bob, she does not need to retrieve Bob's public key from a database. Instead, Alice merely derives the key directly from Bob's identifying information. Databases of public keys are unnecessary. Certificate authorities (“CAs”) also are unnecessary. There is no need to “bind” Bob's identity to his public key because his identity is his public key.

The concept of identity-based cryptosystems is not new. It was proposed in A. Shamir, Identity-Based Cryptosystems and Signatures Schemes, A DVANCES IN C RYPTOGRAPHY— C RYPTO '84, Lecture Notes in Computer Science 196 (1984), Springer, 47-53. However, practical identity-based encryption schemes have not been found until recently. For instance, identity-based schemes were proposed in C. Cocks, An Identity-Based Encryption Scheme Based on Quadratic Residues, available at http://www.cesg.gov.uk/technology/id-pkc/media/ciren.pdf; D. Boneh, M. Franklin, Identity Based Encryption from the Weil Pairing, A DVANCES IN C RYPTOLOGY—CRYPTO 2001, Lecture Notes in Computer Science 2139 (2001), Springer, 213-229; and D. Boneh, M. Franklin, Identity Based Encryption from the Weil Pairing (extended version), available at http://www.cs.stanford.edu/˜dabo/papers/ibe.pdf. Cocks's scheme is based on the “Quadratic Residuosity Problem,” and although encryption and decryption are reasonably fast (about the speed of RSA), there is significant message expansion (i.e., the bit-length of the ciphertext is many times the bit-length of the plaintext). The Boneh-Franklin scheme bases its security on the “Bilinear Diffie-Hellman Problem,” and it is quite fast and efficient when using Weil or Tate pairings on supersingular elliptic curves or abelian varieties.

However, the known identity-based encryption schemes have a significant shortcoming—they are not hierarchical. In non-identity-based public key cryptography, it has been possible to have a hierarchy of CAs in which the root CA can issue certificates for other CAs, who in turn can issue certificates for users in particular domains. This is desirable because it reduces the workload on the root CA. A practical hierarchical scheme for identity-based cryptography has not been developed.

Ideally, a hierarchical identity-based encryption scheme would involve a hierarchy of logical or actual PKGs. For instance, a root PKG may issue private keys to other PKGs, who in turn would issue private keys to users in particular domains. It also would be possible to send an encrypted communication without an online lookup of the recipient's public key or lower-level public parameters, even if the sender is not in the system at all, as long as the sender obtained the public parameters of the root PKG. Another advantage of a hierarchical identity-based encryption scheme would be damage control. For instance, disclosure of a domain PKG's secret would not compromise the secrets of higher-level PKGs, or of any other PKGs that are not direct descendents of the compromised domain PKG. The schemes taught by Cocks and Boneh-Franklin do not have these properties.

A secure and practical hierarchical identity-based encryption scheme has not been developed. A hierarchical identity-based key sharing scheme with partial collusion-resistance is given in G. Hanaoka, T. Nishioka, Y. Zheng, H. Imai, An Efficient Hierarchical Identity-Based Key-Sharing Method Resistant Against Collusion Attacks, A DVANCES IN C RYPTOGRAPHY —A SIACRYPT 1999, Lecture Notes in Computer Science 1716 (1999), Springer 348-362; and G. Hanaoka, T. Nishioka, Y. Zheng, H. Imai, A Hierarchical Non-Interactive Key-Sharing Scheme With Low Memory Size and High Resistance Against Collusion Attacks, to appear in THE COMPUTER JOURNAL. In addition, an introduction to hierarchical identity-based encryption was provided in J. Horwitz, B. Lynn, Toward Hierarchical Identity-Based Encryption, to appear in ADVANCES IN CRYPTOGRAPHY—EUROCRYPT 2002, Lecture Notes in Computer Science. Springer. Horwitz and Lynn proposed a two-level hierarchical scheme with total collusion-resistance at the first level and partial collusion-resistance at the second level (i.e., users can collude to obtain the secret of their domain PKG and thereafter masquerade as that domain PKG). However, the complexity of the Horwitz-Lynn system increases with the collusion-resistance at the second level, and therefore that scheme cannot be both practical and secure.

Accordingly, there has been a need for a secure and practical hierarchical identity-based encryption scheme. It is therefore an object of the present invention to provide a secure and practical hierarchical identity-based encryption scheme. It is another object of the present invention to provide a secure and practical hierarchical identity-based signature scheme. It is a further object of the present invention that the encryption and signature schemes be fully scalable. It is a still further object of the present invention that the encryption and signature schemes have total collusion resistance on an arbitrary number of levels, and that they have chosen-ciphertext security in the random oracle model.

BRIEF SUMMARY OF THE PREFERRED EMBODIMENTS

In accordance with the present invention, methods are provided for implementing secure and practical hierarchical identity-based encryption and signature schemes.

According to one aspect of the present invention, a method is provided for encoding and decoding a digital message between a sender and a recipient in a system including a plurality of private key generators (“PKGs”). The PKGs include at least a root PKG and n lower-level PKG in the hierarchy between the root PKG and the recipient, wherein n≧1. A root key generation secret is selected and is known only to the root PKG. A root key generation parameter is generated based on the root key generation secret. A lower-level key generation secret is selected for each of the n lower-level PKGs, wherein each lower-level key generation secret is known only to its associated lower-level PKG. A lower-level key generation parameter also is generated for each of the n lower-level PKGs using at least the lower-level key generation secret for its associated lower-level private key generator. The message is encoded to form a ciphertext using at least the root key generation parameter and recipient identity information. A recipient private key is generated such that the recipient private key is related to at least the root key generation secret, one or more of the n lower-level key generation secrets associated with the n lower-level PKGs in the hierarchy between the root PKG and the recipient, and the recipient identity information. The ciphertext is decoded to recover the message using at least the recipient private key.

According to another aspect of the present invention, a method is provided for encoding and decoding a digital message between a sender and a recipient in a system including a plurality of private key generators (“PKGs”). The PKGs include at least a root PKG, m lower-level PKGs in the hierarchy between the root PKG and the sender, wherein m≧1, n lower-level PKG in the hierarchy between the root PKG and the recipient, wherein n≧1, and PKG_(l), which is a common ancestor PKG to both the sender and the recipient. In the hierarchy, l of the m private key generators are common ancestors to both the sender and the recipient, wherein l≧1.

According to this aspect of the invention, a lower-level key generation secret is selected for each of the m lower-level PKGs in the hierarchy between the root PKG and the sender. A sender private key is generated such that the sender private key is related to at least the root key generation secret, one or more of the m lower-level key generation secrets associated with the m lower-level PKGs in the hierarchy between the root PKG and the sender, and sender identity information. A recipient private key is generated such that the recipient private key is related to at least the root key generation secret, one or more of the n lower-level key generation secrets associated with the n lower-level PKGs in the hierarchy between the root PKG and the recipient, and recipient identity information. The message is encoded using at least the recipient identity information, the sender private key, and zero or more of the lower-level key generation parameters associated with the (m−l+1) private key generators at or below the level of the common ancestor PKG_(l), but not using any of the lower-level key generation parameters that are associated with the (l−1) PKGs above the common ancestor PKG_(l). The message is decoded using at least the sender identity information, the recipient private key, and zero or more of the lower-level key generation parameters associated with the (n−l+1) private key generators at or below the level of the common ancestor PKG,, but not using any of the lower-level key generation parameters that are associated with the (l−1) PKGs above the common ancestor PKG_(l).

According to another aspect of the present invention, a method is provided for generating and verifying a digital signature of a message between a sender and a recipient in a system including a plurality of PKGs. The PKGs include at least a root PKG and n lower-level PKG in the hierarchy between the root PKG and the sender, wherein n≧1. A root key generation secret is selected and is known only to the root PKG. A root key generation parameter is generated based on the root key generation secret. A lower-level key generation secret is selected for each of the n lower-level PKGs, wherein each lower-level key generation secret is known only to its associated lower-level PKG. A lower-level key generation parameter also is generated for each of the n lower-level PKGs using at least the lower-level key generation secret for its associated lower-level private key generator. A private key is generated for the sender such that the private key is related to at least the root key generation secret and sender identity information. The message is signed to generate the digital signature using at least the sender private key. The digital message is verified using at least the root key generation parameter and the sender identity information.

BRIEF DESCRIPTION OF THE DRAWINGS

The subsequent description of the preferred embodiments of the present invention refers to the attached drawings, wherein:

FIG. 1 shows a flow diagram illustrating a method of encoding and decoding a digital message according to one presently preferred embodiment of the invention;

FIG. 2 shows a flow diagram illustrating a method of encoding and decoding a digital message between a sender y and a recipient z according to another presently preferred embodiment of the invention;

FIG. 3 shows a block diagram illustrating a typical hierarchical structure in which this method of FIG. 2 may be performed;

FIG. 4 shows a flow diagram illustrating a method of encoding and decoding a digital message M communicated between a sender y and a recipient z according to another presently preferred embodiment of the invention;

FIG. 5 shows a flow diagram illustrating a method of encoding and decoding a digital message M communicated between a sender y and a recipient z according to another presently preferred embodiment of the invention;

FIG. 6 shows a flow diagram illustrating a method of encoding and decoding a digital message M communicated between a sender y and a recipient z according to another presently preferred embodiment of the invention;

FIG. 7 shows a flow diagram illustrating a method of generating and verifying a digital signature according to another presently preferred embodiment of the invention;

FIG. 8 shows a flow diagram illustrating a method of generating and verifying a digital signature Sig of a digital message M communicated between a sender y and a recipient z according to another presently preferred embodiment of the invention; and

FIG. 9 shows a flow diagram illustrating a method of generating and verifying a digital signature Sig of a digital message M communicated between a sender y and a recipient z according to another presently preferred embodiment of the invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The presently preferred methods of the invention provide secure and practical hierarchical identity-based encryption (“HIDE”) and signature (“HIDS”) schemes. The hierarchical schemes are fully scalable, have total collusion resistance on an arbitrary number of levels, and have chosen-ciphertext security in the random oracle model. These objectives are achieved, in part, by introducing additional random information at each of the lower-level PKGs. One intuitively surprising aspect of these schemes is that, even though lower level PKGs generate additional random information, this does not necessitate adding public parameters below the root level of the hierarchy. In addition, the random information generated by a lower-level PKG does not adversely affect the ability of users not under the lower-level PKG to send encrypted communications to users under the lower-level PKG.

Each of the HIDE and HIDS schemes of the present invention requires a hierarchical structure of PKGs, including at least one root PKG and a plurality of lower-level PKGs. The hierarchy and the lower-level PKGs may be logical or actual. For instance, a single entity may generate both a root key generation secret and the lower-level key generation secrets from which lower-level users' encryption or signature keys are generated. In this case, the lower-level PKGs are not separate entities, but are merely processes or information arranged in a logical hierarchy and used to generate keys for descendent PKGs and users in the hierarchy. Alternatively, each lower-level PKG may be a separate entity. Another alternative involves a hybrid of actual and logical lower-level PKGs. For purposes of this disclosure, the term “lower-level PKG” will be used generically to refer to any of these alternatives.

In the context of the hierarchical identity-based cryptosystems disclosed herein, identity-based public keys may be based on time periods. For instance, a particular recipient's identity may change with each succeeding time period. Alternatively, a recipient may arrange the time periods as children or descendents of itself in a hierarchy, and a sender would use the identity of the proper time period when encoding the message. Either way, each key may be valid for encrypting messages to Bob only during the associated time period.

The HIDE schemes of the present invention generally include five randomized algorithms: Root Setup, Lower-level Setup, Extraction, Encryption, and Decryption. Three of these algorithms rely upon the identities of the relevant entities in the hierarchy. Each user preferably has a position in the hierarchy that may be defined by its tuple of IDs: (ID₁, . . . , ID_(t)). The user's ancestors in the hierarchy are the root PKG and the users, or PKGs, whose ID-tuples are {(ID₁, . . . , ID_(i)): 1≦i≦(t−1)}. The ID-tuples preferably are represented as binary strings for purposes of computations.

In the Root Setup algorithm, the root PKG uses a security parameter k to generate public system parameters params and a root key generation secret. The system parameters include a description of the message space M and the ciphertext space C. The system parameters will be publicly available, while only the root PKG will know the root key generation secret.

In the Lower-level Setup algorithm, each lower-level PKG preferably generates its own lower-level key generation secret for purposes of extraction. Alternatively, a lower-level PKG may generate random one-time secrets for each extraction.

In the Extraction algorithm, a PKG (whether the root PKG or a lower-level PKG) generates a private key for any of its children. The private key is generated using the system parameters, the generating PKG's private key, and any other preferred secret information.

In the Encryption algorithm, a sender receives the system parameters from the root PKG, preferably via some secure means outside the present system. It is not necessary for the sender to receive any of the lower-level key generation parameters. The sender encodes a message M ε M to generate a ciphertext C ε C using params and the ID-tuple of the intended recipient. Conversely, in the Decryption algorithm, the recipient decodes the ciphertext C to recover the message M using params and the recipient's private key d. Encryption and decryption preferably satisfy the standard consistency constraint:

∀M M ε M: Decryption(params, d, C)=M

where C=Encryption(params, ID-tuple, M).

Like the HIDE schemes, the HIDS schemes of the present invention also generally include five randomized algorithms: Root Setup, Lower-level Setup, Extraction, Signing, and Verification. For Root Setup, the system parameters are supplemented to include a description of the signature space S. Lower-level Setup and Extraction preferably are the same as for HIDE, as described above.

In the Signing algorithm, the sender of a digital message signs the message M ε M to generate a signature S ε S using params and the sender's private key d. In the Verification algorithm, the recipient of the signed message verifies the signature S using params and the ID-tuple of the sender. The Verification algorithm preferably outputs “valid” or “invalid”. Signing and Verification also preferably satisfies a consistency constraint:

∀M ε M: Verification (params, ID-tuple, S)=“valid”

where S=Signing(params, d, M).

Security of HIDE and HIDS Schemes

The security of the schemes embodying the present invention will now be discussed with respect to both HIDE and HIDS. It has been noted in the context of non-hierarchical identity-based cryptography that the standard definition of chosen-ciphertext security must be strengthened for identity-based systems. This is because it should be assumed, for purposes of a security analysis, that an adversary can obtain the private key associated with any identity of its choice (other than the particular identity being attacked). The same applies to hierarchical identity-based cryptography. Accordingly, to establish that the HIDE schemes of the present invention are chosen-ciphertext secure, a simulated attacker is allowed to make private key extraction queries. Also, the simulated adversary is allowed to choose the identity on which it wishes to be challenged.

It should also be noted that an adversary may choose the identity of its target adaptively or nonadaptively. An adversary that chooses its target adaptively will first make hash queries and extraction queries, and then choose its target based on the results of these queries. Such an adversary might not have a particular target in mind when it begins the attack. Rather, the adversary is successful it is able to hack somebody. A nonadaptive adversary, on the other hand, chooses its target independently from results of hash queries and extraction queries. For example, such an adversary might target a personal enemy. The adversary may still make hash queries and extraction queries, but its target choice is based strictly on the target's identity, not on the query results. Obviously, security against an adaptively-chosen-target adversary is the stronger, and therefore preferable, notion of security. However, the security analysis of the HIDE schemes in the present invention address both types of security.

A HIDE scheme is said to be semantically secure against adaptive chosen ciphertext and adaptive chosen target attack if no polynomially bounded adversary A has a non-negligible advantage against the challenger in the following game.

SETUP: The challenger takes a security parameter k and runs the Root Setup algorithm. It gives the adversary the resulting system parameters params. It keeps the root key generation secret to itself.

PHASE 1: The adversary issues queries q₁, . . . , q_(m), where q_(i) is one of:

-   -   1. Public-key query (ID-tuple_(i)): The challenger runs a hash         algorithm on ID-tuple_(i) to obtain the public key H         (ID-tuple_(i)) corresponding to ID-tuple_(i).     -   2. Extraction query (ID-tuple_(i)): The challenger runs the         Extraction algorithm to generate the private key d_(i)         corresponding to ID-tuple_(i), and sends d_(i) to the adversary.     -   3. Decryption query (ID-tuple_(i), C_(i)): The challenger runs         the Extraction algorithm to generate the private key d_(i)         corresponding to ID-tuple_(i), runs the Decryption algorithm to         decrypt C_(i) using d_(i), and sends the resulting plaintext to         the adversary.         These queries may be asked adaptively. In addition, the queried         ID-tuple_(i) may correspond to a position at any level of the         hierarchy.

CHALLENGE: Once the adversary decides that Phase 1 is over, it outputs two equal-length plaintexts M₀, M₁ ε M and an ID-tuple on which it wishes to be challenged. The only constraints are that neither this ID-tuple nor its ancestors appear in any private key extraction query in Phase 1. The challenger picks a random bit b ε {0,1} and sets C=Encryption(params, ID-tuple, M_(b)). It sends C as a challenge to the adversary.

PHASE 2: The adversary issues more queries q_(m+1), . . . , q_(n) where q_(i) is one of:

-   -   1. Public-key query (ID-tuple_(i)): The challenger responds as         in Phase 1.     -   2. Extraction query (ID-tuple_(i)): The challenger responds as         in Phase 1.     -   3. Decryption query (C, ID-tuple_(i)): The challenger responds         as in Phase 1.         The queries in Phase 2 are subject to the constraint that the         challenger cannot make an Extraction query on the ID-tuple         associated with the challenge ciphertext C, or make a Decryption         query using that ID-tuple and the ciphertext C. This same         constraint also applies to all ancestors of the ID-tuple.

GUESS: The adversary outputs a guess b′ ε{0,1}. The adversary wins the game if b=b′. The adversary's advantage in attacking the scheme is defined to be |Pr[b=b′]−½|.

A HIDE schemes is said to be a one-way encryption scheme if no polynomial time adversary has a non-negligible advantage in the game described below. In this game, the adversary A is given a random public key K_(pub) and a ciphertext C that is the encryption of a random message Musing K_(pub), and outputs a guess for the plaintext. The adversary is said to have an advantage ε against the scheme if ε is the probability that A outputs M. The game is played as follows:

SETUP: The challenger takes a security parameter k and runs the Root Setup algorithm. It gives the adversary the resulting system parameters params. It keeps the root key generation secret to itself.

PHASE 1: The adversary makes public key and/or extraction queries as in Phase 1 of the chosen-ciphertext security analysis described above.

CHALLENGE: Once the adversary decides that Phase 1 is over, it outputs a new ID-tuple ID on which it wishes to be challenged. The challenger picks a random M ε M and sets C=Encryption(params, ID-tuple, M). It sends C as a challenge to the adversary.

PHASE 2: The adversary issues more public-key queries and more extraction queries on identities other than ID and its ancestors, and the challenger responds as in Phase 1.

GUESS: The adversary outputs a guess M′ ε M. The adversary wins the game if M=M′. The adversary's advantage in attacking the scheme is defined to be Pr[M=M′].

The schemes of the present invention are secure against the challenges described above. In addition, the HIDS schemes of the present invention are secure against existential forgery on adaptively chosen messages. An adversary should be unable to forge its target's signature on other messages that the target has not signed previously, even after (adaptively) obtaining the target's signature on messages of the adversary's choosing. A HIDS adversary also will have the ability to make public key queries and private key extraction queries on entities other than the target and its ancestors, and the ability to choose its target. As with HIDE, the adversary's choice of target may be adaptive or nonadaptive.

Pairings

The presently preferred HIDE and HIDS schemes of the present invention are based on pairings, such as, for instance, the Weil or Tate pairings associated with elliptic curves or abelian varieties. The methods also are based on the Bilinear Diffie-Hellman problem. They use two cyclic groups G₁ and G₂, preferably of the same large prime order q. The first group G₁ preferably is a group of points on an elliptic curve or abelian variety, and the group law on G₁ may be written additively. The second group G₂ preferably is a multiplicative subgroup of a finite field, and the group law on G₂ may be written multiplicatively. However, other types of groups may be used as G₁ and G₂ consistent with the present invention.

The methods also use a generator P₀ of the first group G₁. In addition, a pairing or function ê:G₁×G₁→G₂ is provided for mapping two elements of the first group G₁ to one element of the second group G₂. The function ê preferably satisfies three conditions. First, the function ê preferably is bilinear, such that if Q and R are in G₁ and a and b are integers, then ê(aQ, bR)=ê(Q, R)^(ab). Second, the function ê preferably is non-degenerate, such that the map does not send all pairs in G₁×G₁ to the identity in G₂. Third, the function ê preferably is efficiently computable. A function ê satisfying these three conditions is considered to be admissible.

The function ê also preferably is symmetric, such that ê(Q, R)=ê(R, Q) for all Q, R ε G₁. Symmetry, however, follows immediately from the bilinearity and the fact that G₁ is a cyclic group. Weil and Tate pairings associated with supersingular elliptic curves or abelian varieties can be modified to create such bilinear maps according to methods known in the art. However, even though reference to elements of the first cyclic group G₁ as “points” may suggest that the function ê is a modified Weil or Tate pairing, it should be noted that any admissible pairing e will work.

The security of the HIDE and HIDS schemes of the present invention is based primarily on the difficulty of the Bilinear Diffie-Hellman problem. The Bilinear Diffie-Hellman problem is that of finding ê(P, p)^(abc) given a randomly chosen P ε G₁, as well as aP, bP, and cP (for unknown randomly chosen a, b, c ε Z/qZ). Solving the Diffie-Hellman problem in G₁ solves the Bilinear Diffie-Hellman problem because ê(P, p)^(abc)=ê(abP, cP). Similarly, solving the Diffie-Hellman problem in G₂ solves the Bilinear Diffie-Hellman problem because, if g=ê(P, P), then g^(abc)=(g^(ab))^(c) where g^(ab)=ê(aP, bP) and g^(c)=ê(P, cP). For the Bilinear Diffie-Hellman problem to be hard, G₁ and G₂ should be chosen such that there is no known algorithm for efficiently solving the Diffie-Hellman problem in either G₁ or G₂. If the Bilinear Diffie-Hellman problem is hard for a pairing ê, then it follows that e is non-degenerate.

A randomized algorithm IG is a Bilinear Diffie-Hellman generator if IG takes a security parameter k>0, runs in time polynomial in k, and outputs the description of two groups G₁ and G_(2,) preferably of the same prime order q, and the description of an admissible pairing ê:G₁×G₁→G₂. If IG is a Bilinear Diffie-Hellman parameter generator, the advantage Adv_(IG)(B) that an algorithm B has in solving the Bilinear Diffie-Hellman problem is defined to be the probability that the algorithm B outputs ê(P, P)^(abc) when the inputs to the algorithm are G₁, G₂, ê, P, aP, bP, and cP, where (G₁, G₂, ê) is the output of IG for a sufficiently large security parameter k, P is a random generator of G₁, and a, b, and c are random elements of Z/qZ. The assumption underlying the Bilinear Diffie-Hellman problem is that Adv_(IG)(B) is negligible for all efficient algorithms B.

HIDE Schemes

Referring now to the accompanying drawings, FIG. 1 shows a flow diagram illustrating a method of encoding and decoding a digital message according to one presently preferred embodiment of the invention. The method is performed in a HIDE system including a plurality of PKGs. The PKGs include at least a root PKG and n lower-level PKGs in the hierarchy between the root PKG and the recipient, wherein n≧1.

In block 102, the root PKG selects a root key generation secret known only to the root PKG. The root key generation secret may be used to generate private keys for PKGs and/or users below the root PKG in the hierarchy. The root PKG then generates a root key generation parameter based on the root key generation secret in block 104. The root key generation parameter is used to mask the root key generation secret. The root key generation parameter may be revealed to lower-level PKGs without compromising the root key generation secret. The lower-level PKGs select lower-level key generation secrets in block 106. The lower-level key generation secret associated with a given lower-level PKG may be used to generate private keys for PKGs and/or users below the associated lower-level PKG in the hierarchy. Like the root key generation secret, each of the lower-level key generation secrets is known only to its associated lower-level PKG.

In block 108, lower-level key generation parameters are generated for each of the n lower-level PKGs. Each of the lower-level key generation parameters is generated using at least the lower-level key generation secret for its associated lower-level PKG. Like the root key generation parameter, each of the lower-level key generation parameters masks its associated lower-level key generation secret.

Using at least the root key generation parameter and identity information associated with the recipient, the sender encodes the message in block 110 to form a ciphertext. For instance, the message may be encoded using only the root key generation parameter and the recipient's identity. Alternatively, one of the lower-level key generation parameters may be used, such as is described in more detail below with respect to dual-HIDE schemes. In block 112, a lower-level PKG generates a private key for the recipient such that the private key is related to at least the root key generation secret, one or more of the n lower-level key generation secrets associated with the n lower-level PKGs in the hierarchy between the root PKG and the recipient, and the recipient's identity information. For instance, in addition to root key generation secret and the recipient's identity information, the recipient's private key preferably also is related at least to the lower-level key generation secret of the PKG that issued the private key to the recipient. Alternatively, the recipient's private key may be related to all n of its ancestral PKG's lower-level key generation secrets, as well as the root key generation secret. In block 114, the recipient uses at least its private key to decode the ciphertext and recover the message. In addition to using its private key to decode, the recipient preferably also uses the n lower-level key generation parameters associated with the n lower-level PKGs in the hierarchy between the root PKG and the recipient.

Each lower-level PKG has a key generation secret, just like the root PKG. As described above, a lower-level PKG preferably uses this secret to generate a private key for each of its children, just as the root PKG does. As a result, the children's private keys are related to the lower-level PKG's key generation secret. This is true even if the lower-level PKG uses a modified version of its key generation secret to obscure that secret for purposes of restricting key escrow, as described more fully below. At the same time, the lower-level PKGs need not always use the same secret for each private key extraction. Rather, a new key generation secret could be generated randomly for each of the PKG's children, resulting in a different key generation parameter for each child.

Because a lower-level PKG is able to generate a private key for the recipient (block 112), the root PKG need not generate all of the private keys itself. In addition, because the lower-level PKGs use their own key generation secrets to generate private keys for their descendants, compromising a lower-level key generation secret causes only limited security damage to the hierarchy. Rather than compromising all of the private keys in the hierarchy, a breach of a lower-level PKG compromises only the private key of that PKG and those private keys that were generated using that PKG's key generation secret (i.e., the private keys of those users that are direct hierarchical descendants of the compromised PKG).

Another advantage of this embodiment is that the sender need not be in the hierarchy to send an encoded message to the recipient. The sender merely needs to know the identity information associated with the recipient and the system parameters generated by the root PKG. There are however, certain additional advantages of the HIDE schemes of the present invention that become available when the sender is positioned within the hierarchy. For instance, when both the sender and the recipient are in the hierarchy, the efficiency of the message encryption may be improved by using the identities of both parties. This type of HIDE scheme may be referred to as dual-HIDE because the identities of both the sender and the recipient are used as input for the encryption and decryption algorithms. A method of encoding and decoding a message using a dual-HIDE scheme will now be discussed with reference to FIGS. 2 and 3.

Dual-HIDE

FIG. 2 shows a flow diagram illustrating a method of encoding and decoding a digital message between a sender y and a recipient z according to another presently preferred embodiment of the invention. FIG. 3 shows a block diagram illustrating a typical hierarchical structure in which this method may be performed. Like the previous embodiment, this method is performed in a HIDE system including at least a root PKG 302 and n lower-level PKGs 304 a,b,d in the hierarchy between the root PKG 302 and the recipient z 308, wherein n≧1. The sender y 306 in this embodiment also must be in the hierarchy, and the hierarchy also includes m lower-level PKGs 304 a,b,c between the root PKG 302 and the sender y 306, wherein m≧1. Of the m PKGs 304 a,b,c between the root PKG 302 and the sender y 306, and the n PKGs 304 a,b,d between the root PKG 302 and the recipient z 308, there are l PKGs 304 a,b that are common ancestors to both the sender y 306 and the recipient z 308, wherein 1≦l≦m, n. For instance, two of these l common ancestral PKGs (PKG_(y1)/PKG_(z1) 304 a and PKG_(yl)/PKG_(zl) 304 b) are shown in FIG. 3.

The method of this embodiment begins in block 202, when the root PKG 302 selects a root key generation secret known only to the root PKG 302. The root PKG 302 then generates a root key generation parameter based on the root key generation secret in block 204. The lower-level PKGs 304 a-d select lower-level key generation secrets in block 206. Like the root key generation secret, each of the lower-level key generation secrets is known only to its associated lower-level PKG 304 a-d. In block 208, lower-level key generation parameters are generated for each of the n lower-level PKGs 304 a-d. Each of the lower-level key generation parameters is generated using at least the lower-level key generation secret for its associated lower-level PKG 304 a-d.

In block 210, the sender's parent PKG_(ym) 304 c generates a private key for the sender y 306 such that the private key is related to at least the root key generation secret, one or more of the m lower-level key generation secrets associated with the m lower-level PKGs 304 a,b,c between the root PKG 302 and the sender y 306, and the sender's identity information. For instance, in addition to root key generation secret and the sender's identity information, the sender's private key preferably is related at least to the lower-level key generation secret of the sender's parent PKG_(ym) 304 c. Alternatively, the sender's private key may be related to all m of its direct ancestral PKGs' lower-level key generation secrets, as well as the root key generation secret. In block 212, the recipient's parent PKG_(zn) 304 d generates a private key for the recipient z in a similar manner that the sender's parent PKG_(ym) 304 c used to generate the sender's private key.

In block 214, the sender y encodes the message to form a ciphertext using at least the sender's private key and one or more of the lower-level key generation parameters associated with the (m−l+1) PKGs (i.e., PKG_(yl), 304 b and PKG_(ym) 304 c) between the root PKG 302 and the sender y 306 that are at or below the level of the lowest ancestor PKG (PKG_(yl)/PKG_(zl) 304 b) that is common to both the sender y 306 and the recipient z 308. In encoding the message, the sender y 306 preferably does not use any of the lower-level key generation parameters that are associated with the (l−1) PKGs (i.e., PKG_(y1) 304 a) that are above the lowest common ancestor PKG (PKG_(yl)/PKG_(zl) 304 b).

The recipient z 308 then decodes the ciphertext to recover the message in block 216 using at least the recipient's private key and one or more of the lower-level key generation parameters associated with the (n−l+1) PKGs (i.e., PKG_(zl), 304 b and PKG_(zn) 304 c) between the root PKG 302 and the recipient z 308 that are at or below the level of the lowest ancestor PKG (PKG_(yl)/PKG_(zl) 304 b) that is common to both the sender y 306 and the recipient z 308. In decoding the message, the recipient z 306 preferably does not use any of the lower-level key generation parameters that are associated with the (l−1) PKGs (i.e., PKG_(z1) 304 a) that are above the lowest common ancestor PKG (PKG_(yl)/PKG_(zl) 304 b).

This dual-HIDE embodiment of the invention provides a more efficient scheme for encoding and decoding the message because it requires the use of fewer key generation parameters. For instance, decoding in a regular HIDE scheme preferably requires all n of the key generation parameters, but decoding in a dual-HIDE scheme preferably requires only (n−l+1) of the key generation parameters. Dual-HIDE schemes require the sender y 306 to obtain its private key before sending an encoded message to the recipient z 308, as opposed to merely obtaining the public system parameters of the root PKG. The dual-HIDE schemes also enable the sender y 306 and the recipient z 308 to restrict the scope of key escrow, as described more fully below. This shared secret is unknown to third parties other than their lowest common ancestor PKG_(yl)/PKG_(zl) 304 b.

BasicHIDE

In some embodiments, the scheme is as follows. Let Level_(i) be the set of entities at level i, where Level₀={Root PKG}. Let K be the security parameter given to the setup algorithm and let IG be a BDH parameter generator.

Root Setup: The root PKG:

-   -   1. runs IG on input K to generate groups G₁, G₂ of some prime         order q and an admissible pairing ê: G₁×G₁→G₂;     -   2. chooses an arbitrary generator P₀εG₁;     -   3. picks a random s₀εZ/qZ and sets Q₀=s₀P₀;     -   4. chooses cryptographic hash functions H₁:{0,1}*→G₁ and         H₂:G₂→{0,1}^(n) for some n. The security analysis will treat H₁         and H₂ as random oracles.

The message space is M={0,1}^(n). The ciphertext space C=G₁ ^(t)×{0,1}^(t) where t is the level of the recipient. The system parameters are params={G₁,G₂,ê,P₀,Q₀,H₁,H₂}. The root PKG's secret is s₀εZ/qZ.

Lower-level Setup. Entity E_(t)εLevel_(t) picks a random s_(t)εZ/qZ, which it keeps secret.

Extraction: Let E_(t) be an entity in Level_(t) with ID-tuple (ID₁, . . . , ID_(t)), where (ID₁, . . . , ID_(i)) for 1≦i≦t is the ID tuple of E_(t)'s ancestor at Level_(i). Set S₀ to be the identity element of G₁. The E_(t)'s parent:

-   -   1. computes P_(t)=H₁(ID₁, . . . , ID_(t))εG₁;     -   2. sets E_(t)'s secret point S_(t) to be         ${{S_{t - 1} + {s_{t - 1}P_{t}}} = {\sum\limits_{i = 1}^{t}{s_{i - 1}P_{i}}}};$     -   3. also gives E_(t) the values of Q_(i)=s_(i)P₀ for 1≦i≦t−1.

Encryption: To encrypt M ε M with the ID-tuple (ID₁, . . . , ID_(t)), do the following:

-   -   1. Compute P_(i)=H₁(ID₁, . . . , ID_(i))εG₁ for 1≦i≦t.     -   2. Choose a random rεZ/qZ.     -   3. Set the ciphertext to be         C=[rP ₀ ,rP ₂ , . . . , rP _(t) ,M⊕H ₂(g ^(r))] where g=ê(Q ₀ ,P         ₁)εG ₂.

Decryption: Let C=[U₀,U₂, . . . , U_(t),V]εC be the ciphertext encrypted using the ID-tuple (ID₁, . . . , ID_(t)). To decrypt C, E_(t) computes: ${V \oplus {H_{2}\left( \frac{\hat{e}\left( {U_{0},S_{t}} \right)}{\prod\limits_{i = 2}^{t}{\hat{e}\left( {Q_{i - 1},U_{i}} \right)}} \right)}} = {M.}$

This concludes the description of one embodiment of our BasicHIDE scheme.

Remark 1. Each lower-level PKG—say, in Level_(t)—has a secret s_(t)εZ/qZ, just like the root PKG. A lower-level PKG uses this secret to generate a secret point for each of its children, just as the root PKG does. An interesting fact, however, is that lower-level PKGs need not always use the same s_(t) for each private key extraction. Rather, s_(t) could be generated randomly for each of the PKG's children.

Remark 2. H₁ can be chosen to be an iterated hash function so that, for example, P_(i) may be computed as H₁(P_(i−1),ID_(i)) rather than H₁(ID₁, . . . , ID_(i)).

FIG. 4 shows a flow diagram illustrating a method of encoding and decoding a digital message M communicated between a sender y and a recipient z according to another presently preferred embodiment of the invention. The recipient z 308 is n+1 levels below the root PKG in the hierarchy, as shown in FIG. 3, and is associated with the ID-tuple (ID_(z1), . . . , ID_(z(n+1))). The recipient's ID-tuple includes identity information ID_(z(n+1)) associated with the recipient, as well as identity information ID_(zi) associated with each of its n ancestral lower-level PKGs in the hierarchy. The method begins in block 402 by generating first and second cyclic groups G₁ and G₂ of elements. In block 404, a function ê is selected, such that the function ê is capable of generating an element of the second cyclic group G₂ from two elements of the first cyclic group G₁. The function ê preferably is an admissible pairing, as described above. A root generator P₀ of the first cyclic group G₁ is selected in block 406. In block 408, a random root key generation secret s₀ associated with and known only to the root PKG 302 is selected. Preferably, s₀ is an element of the cyclic group Z/qZ. A root key generation parameter Q₀=s₀P₀ is generated in block 410. Preferably, Q₀ is an element of the first cyclic group G₁. In block 412, a first function H₁ is selected such that H₁ is capable of generating an element of the first cyclic group G₁ from a first string of binary digits. A second function H₂ is selected in block 414, such that H₂ is capable of generating a second string of binary digits from an element of the second cyclic group G₂. The functions of blocks 402 through 414 are part of the HIDE Root Setup algorithm described above, and preferably are performed at about the same time. By way of example, the functions such as those disclosed in Boneh-Franklin may be used as H₁ and H₂.

The next series of blocks (blocks 416 through 424) show the functions performed as part of Lower-level Setup algorithm. In block 416, a public element P_(zi) is generated for each of the recipients' n ancestral lower-level PKGs. Each of the public elements, P_(zi)=H₁(ID₁, . . . , ID_(zi)) for 1≦i≦n, preferably is an element of the first cyclic group G₁. Although represented in a single block, generation of all the public elements P_(zi) may take place over time, rather than all at once.

A lower-level key generation secret s_(zi) is selected (block 418) for each of the recipients' n ancestral lower-level PKGs 304 a,b,d. The lower-level key generation secrets s_(zi) preferably are elements of the cyclic group Z/qZ for 1≦i≦n, and each lower-level key generation secret s_(zi) preferably is known only to its associated lower-level PKG. Again, although represented in a single block, selection of all the lower-level key generation secrets s_(zi) may take place over time, rather than all at once.

A lower-level secret element S_(zi) is generated (block 420) for each of the sender's n ancestral lower-level PKGs. Each lower-level secret element, S_(zi)=S_(z(i−1))+s_(z(i−1))P_(zi) for 1≦i≦n preferably is an element of the first cyclic group G₁. Although represented in a single block like the public elements P_(zi) and the secrets s_(zi), generation of all the secret elements S_(zi) may take place over time, rather than all at once. For purposes of these iterative key generation processes, S₀ may be defined to be the identity element of G₁.

A lower-level key generation parameter Q_(zi) also is generated (block 422) for each of the recipients' n ancestral lower-level PKGs. Each of the key generation parameters, Q_(zi)=s_(zi)P₀ for 1≦i≦n, preferably is an element of the first cyclic group G₁. Again, although represented in a single block, generation of all the key generation parameters Q_(zi) may take place over time, rather than all at once.

The functions of the next two blocks (blocks 424 and 426) are performed as part of the Extraction algorithm described above. A recipient public element P_(z(n+1)) associated with the recipient z is generated in block 424. The recipient public element, P_(z(n+1))=H₁(ID_(z1), . . . , ID_(z(n+1))), preferably is an element of the first cyclic group G₁. A recipient secret element S_(z(n+1)) associated with the recipient z is then generated in block 426. The recipient secret element ${S_{z{({n + 1})}} = {{S_{zn} + {s_{zn}P_{z{({n + 1})}}}} = {\sum\limits_{i = 1}^{n + 1}{S_{z{({i - 1})}}P_{zi}}}}},$ also preferably is an element of the first cyclic group G₁.

For convenience, the first function H₁ optionally may be chosen to be an iterated function so that, for example, the public points P_(i) may be computed as H₁(P_(z(i−1)), ID_(zi)) rather than H₁(ID₁, . . . , ID_(zi)).

The last two blocks shown in FIG. 4 (blocks 428 and 430) represent the Encryption and Decryption algorithms described above. In block 428, the message M is encoded to generate a ciphertext C. The encoding preferably uses at least the root key generation parameter Q₀ and the ID-tuple (ID_(z1), . . . , ID_(z(n+1))). The ciphertext C is then decoded in block 430 to recover the message M. The decoding preferably uses at least the lower-level key generation parameters Q_(zi) for 1≦i≦n, and the recipient secret element S_(z(n+1)).

The blocks shown in FIG. 4 need not all occur in sequence. For instance, a sender who knows a recipient's identity may encrypt communications to the recipient before the recipient obtains its private key.

The specific use of the parameters and elements described above in the encoding and decoding of the message M and the ciphertext C will now be discussed with reference to FIGS. 5 and 6. FIG. 5 shows a flow diagram illustrating a method of encoding and decoding a digital message M communicated between a sender y and a recipient z according to another presently preferred embodiment of the invention. In this scheme, which may be referred to as BasicHIDE, the Root Setup, Lower-level Setup, and Extraction algorithms are the same as for the embodiment shown in blocks 402 through 426 of FIG. 4. The flow diagram of FIG. 5 illustrates the Encryption and Decryption algorithms, beginning with the selection of a random encryption parameter r in block 528 a. Preferably, r is an integer of the cyclic group Z/qZ. The ciphertext C is then generated in block 528 b using the formula C=[U₀, U₂, . . . , U_(n+1), V]. The ciphertext C includes elements U_(i)=rP_(zi) for i=0 and for 2≦i≦n+1, which relate to the location of the recipient in the hierarchy. The other part of the ciphertext C is the actual message in encrypted form, V=M ⊕ H₂(g^(r)), wherein g=ê(Q₀, P_(z1)). The element g preferably is a member of the second cyclic group G₂. After the message has been encoded, it may be decoded according to the BasicHIDE Decryption algorithm, in which the message M is recovered from the ciphertext C (block 530) using the formula $M = {V \oplus {{H_{2}\left( \frac{\hat{e}\left( {U_{0},S_{n + 1}} \right)}{\prod\limits_{i = 2}^{n + 1}{\hat{e}\left( {Q_{i - 1},U_{i}} \right)}} \right)}.}}$ FullHIDE

It is known that Fujisaki-Okamato padding can be used to convert a basic IBE scheme to an IBE scheme that is chosen ciphertext secure in the random oracle model. In the same way, BasicHIDE can be converted to FullHIDE, a HIDE scheme that is chosen ciphertext secure in the random oracle model. Next we describe the scheme FullHIDE. One embodiment is as follows:

Setup: As in the BasicHIDE scheme, but in addition choose hash functions H₃:{0,1}^(n)×{0,1}^(n)→Z/qZ and H₄:{0,1}^(n)→{0,1}^(n).

Extraction: As in the BasicHIDE scheme.

Encryption: To encrypt M ε M with the ID-tuple (ID₁, . . . , ID_(t)), do the following:

-   -   1. compute P_(i)=H₁(ID₁, . . . , ID_(i))ε=G₁ for 1≦i≦t,     -   2. choose a random σε{0,1}^(n),     -   3. set r=H₃(σ,M), and     -   4. set the ciphertext to be         C=[rP ₀ ,rP ₂ , . . . , rP _(t) ,σ⊕H ₂(g ^(r)), M ⊕ H ₄(σ)]         where g=ê(Q₀,P₁)ε G₂ as before.

Decryption: Let C=[U₀,U₂, . . . , U_(t), V, W]εC be the ciphertext encrypted using the ID-tuple (ID₁, . . , ID_(t)). If (U₀,U₂, . . . , U_(t))ø G₁ ^(t), reject the ciphertext. To decrypt C, E_(t) does the following:

-   -   1. computes         ${{V \oplus {H_{2}\left( \frac{\hat{e}\left( {U_{0},S_{t}} \right)}{\prod\limits_{i = 2}^{t}{\hat{e}\left( {Q_{i - 1},U_{i}} \right)}} \right)}} = \sigma},$     -   2. computes W ⊕ H₄(σ)=M,     -   3. sets r=H₃(σ,M) and tests that U₀=rP₀ and U_(i)=rP_(i) for         i=2, . . . , t. If not, it rejects the ciphertext.     -   4. outputs M as the decryption of C.

Note that M is encrypted as W=M ⊕ H₄(σ). This can be replaced by $W = {{\,\underset{4}{E}}\quad{\,_{H{(\sigma)}}(M)}}$ where E is a semantically secure symmetric encryption scheme.

Using known methods for making one-way encryption schemes secure against chosen-ciphertext attacks, a BasicHIDE scheme may be converted to a FullHIDE scheme that is chosen ciphertext secure in the random oracle model. A FullHIDE scheme that is chosen ciphertext secure will now be discussed with reference to FIG. 6.

FIG. 6 shows a flow diagram illustrating a method of encoding and decoding a digital message M communicated between a sender y and a recipient z according to another presently preferred embodiment of the invention. The Root Setup, Lower-level Setup, and Extraction algorithms are the same for this embodiment of the invention as for the embodiment described with reference to FIG. 4, except that the Root Setup algorithm of this embodiment requires two additional functions. Accordingly, the flow diagram of FIG. 6 begins with the selection of the additional functions (blocks 615 a and 615 b) and continues with the Encryption and Decryption algorithms (blocks 628 a through 630 d).

The Root Setup algorithm is completed by selecting a third function H₃ (block 615 a) and a fourth function H₄ (block 615 b). The third function H₃ preferably is capable of generating an integer of the cyclic group Z/qZ from two strings of binary digits. The fourth function H₄ preferably is capable of generating one binary string from another binary string.

The Encryption algorithm begins with block 628 a, which shows the selection of a random binary string σ. The random binary string σ is then used to generate a random integer r=H₃(σ, M, W), as shown in block 628 b, wherein W is a symmetric encryption of the actual message M. The encryption preferably is generated using a symmetric encryption algorithm E, and using H₄(σ) as the encryption key. Accordingly, W=E_(H) ₄ _((σ))(M). In block 628 c, the ciphertext C=[U₀, U₂, . . . , U_(n+1), V, W] is generated. The ciphertext C includes elements U_(i)=rP_(zi) for i=0 and for 2≦i≦n+1, which relate to the location of the recipient in the hierarchy. The second part of the ciphertext C is the random binary string a in encrypted form, V=σ ⊕ H₂(g^(r)), wherein g=ê(Q₀, P_(z1)). The element g preferably is a member of the second cyclic group G₂. The third part of the ciphertext C is W, the actual message in symmetrically encrypted form, as described above.

The Decryption algorithm begins with block 630 a, which shows the recovery of the random binary string σ. The random binary string σ is recovered using the formula $\sigma = {V \oplus {{H_{2}\left( \frac{\hat{e}\left( {U_{0},S_{z{({n + 1})}}} \right)}{\prod\limits_{i = 2}^{n + 1}{\hat{e}\left( {Q_{i - 1},U_{i}} \right)}} \right)}.}}$ The message M is then recovered from the ciphertext C (block 630 b) using the formula M=E_(H) ₄ _((σ)) ⁻¹(W). The ciphertext optionally may be checked for internal consistency. For instance, an experimental random integer r′=H₃(σ, M, W) may be generated, as shown in block 630 c. The experimental random integer r′ then may be used in block 630 d to confirm that U₀=r′P₀ and U_(i)=r′P_(zi) for 2≧i≧n+1. If so, then the ciphertext C is considered to be authentic. Dual-BasicHIDE and Dual-FullHIDE

In 2000, Sakai, Ohgishi and Kasahara presented a “key sharing scheme” based on the Weil pairing. The idea was quite simple: suppose a PKG has a master secret s, and it issues private keys to users of the form sP_(y) where P_(y)=H₁(ID_(y)) and ID_(y) is the ID of user y (as in Boneh-Franklin). Then users y and z have a shared secret that only they (and the PKG) may compute, namely ê(sP_(y),P_(z))=ê(P_(y),sP_(z)). They may use this shared secret to encrypt their communications. Notice that this “key sharing scheme” does not require any interaction between the parties. We can view Sakai, Ohgishi and Kasahara's discovery as a type of “dual-identity-based encryption,” where the word “dual” indicates that the identities of both the sender and the recipient (rather than just the recipient) are required as input into the encryption and decryption algorithms. The only significant practical difference between this scheme and the Boneh-Franklin IBE scheme is that the sender must obtain its private key from the PKG before sending encrypted communications, as opposed to merely obtaining the public parameters of the PKG.

In the non-hierarchical context, Dual-IBE does not appear to have any substantial advantages over IBE. In the hierarchical context, however, Dual-HIDE may be more efficient than HIDE if the sender and recipient are close to each other in the hierarchy tree. Suppose two users, y and z, have the ID-tuples (ID_(y1), . . . , ID_(yl), . . . , ID_(ym)) and (ID_(z1), . . . , ID_(zl), . . . , ID_(zn)), where (ID_(y1). . . , ID_(yl))=(ID_(z1), . . . , ID_(zl)). In other words, user y is in Level_(m), user z is in Level_(n), and they share a common ancestor in Level_(l). User y may use Dual-HIDE to encrypt a message to user z as follows:

Encryption: To encrypt M ε M, user y:

-   -   1. Computes P_(zi)=H₁(ID_(z1), . . . , ID_(zi))εG₁ for l+1≦i≦n.     -   2. Chooses a random rεZ/qZ.     -   3. Set the ciphertext to be:         C=[rP ₀ ,rP _(z(l+1)) , . . . , rP _(zn) , M⊕H ₂(g _(yl) ^(r))]         where         ${g_{yl} = {\frac{\hat{e}\left( {P_{0},S_{y}} \right)}{\prod\limits_{i = {l + 1}}^{m}{\hat{e}\left( {Q_{y{({i - 1})}},U_{i}} \right)}} = {\hat{e}\left( {P_{0},S_{yl}} \right)}}},$         S_(y) is y's secret point, S_(yl) is the secret point of y's and         z's common ancestor at level l, and         Q_(yi)=s_(yi)P₀ where s_(yi) is the secret number chosen by y's         ancestor at level i.

Decryption: Let C=[U₀,U_(l+1), . . . , U_(n), V] be the ciphertext. To decrypt C, user z computes: ${V \oplus {H_{2}\left( \frac{\hat{e}\left( {U_{0},S_{z}} \right)}{\prod\limits_{i = {l + 1}}^{m}{\hat{e}\left( {Q_{z{({i - 1})}},U_{i}} \right)}} \right)}} = {M.}$

Note that if y and z have a common ancestor below the root PKG, then the ciphertext is shorter with Dual-HIDE than with non-dual HIDE. Further, using Dual-HIDE, the encrypter y computes m−l+1 pairings while the decryptor z computers n−l+1 pairings. (Note that m+n−2l is the “length” of the path between y and z in the hierarchy tree.) In the non-dual HIDE scheme, the encrypter computes one pairing (or receives it as a per-computer value) while the decrypter computers n pairings. Thus when m<2l−1, the total work is less with Dual-HIDE than with non-dual HIDE. The relative computing power of the sender and recipient can also be taken into account.

The number of pairings that y and z must compute can be decreased to m+n−2l+1 if their common ancestor in Level, always uses the same s_(l) rather than generating this number randomly with each private key extraction. Encryption and decryption proceed as follows:

Encryption: To encrypt M ε M, user y:

-   -   1. Computes P_(zi)=H₁(ID_(z1), . . . , ID_(zi))εG₁ for l+1≦i≦n.     -   2. Chooses a random rεZ/qZ.     -   3. Set the ciphertext to be:         C=[rP ₀ ,r(P _(z(l+1)) −P _(z(l+1))),rP _(z(l−2)) , . . . , rP         _(zn) , M⊕H ₂(g _(y(l+1)) ^(r))]         where         ${g_{y{({l + 1})}} = {\frac{\hat{e}\left( {P_{0},S_{y}} \right)}{\prod\limits_{i = {l + 2}}^{m}{\hat{e}\left( {Q_{y{({i - 1})}},P_{yi}} \right)}} = {\hat{e}\left( {P_{0},S_{y{({l + 1})}}} \right)}}},$         where S_(y) is y's secret point and S_(y(l+1)) is the secret         point of y's ancestor at level l+1.

Decryption: Let C=[U₀,U_(l+1), . . . , U_(n),V] be the ciphertext. To decrypt C, user z computes: ${V \oplus {H_{2}\frac{{\hat{e}\left( {U_{0},S_{z}} \right)}{\hat{e}\left( {U_{l + 1},Q_{zl}} \right)}}{\prod\limits_{i = {l + 2}}^{n}\quad{\hat{e}\left( {Q_{z{({i - 1})}},U_{i}} \right)}}}} = {M.}$

The concept of dual-HIDE described with reference to FIGS. 2 and 3 may be applied to BasicHIDE and FullHIDE schemes. When both the sender and recipient are within the hierarchical structure, as shown in FIG. 3, dual-HIDE allows them to increase the efficiency and security of their encrypted communications. The application of dual-HIDE to BasicHIDE and FullHIDE schemes requires the determination of additional information, most of which is determined via the Lower-level Setup algorithm described above. For instance, public elements P_(yi), lower-level key generation secrets s_(yi), lower-level secret elements S_(yi), and lower-level key generation parameters Q_(yi) must be determined for the sender's m ancestral lower-level PKGs. Note, however, that for the lower-level PKGs that are common ancestors to both the sender y and the recipient z, these parameters preferably will be the same for purposes of analyzing both the sender y and the recipient z (i.e., preferably for all i≦l: P_(yi)=P_(zi), s_(yi)=s_(zi), S_(yi)=S_(zi), and Q_(yi)=Q_(zi)) Dual-HIDE also requires determination of a sender public element P_(y(m+1)) and a sender secret element S_(y(m+1)) for the sender, using the same methods for which these parameters are determined for the recipient as described above.

Given these additional parameters, a message M may be encoded to generate a ciphertext C according the principles of dual-HIDE by using the lower-level key generation parameters Q_(yi) for i≧l and the sender secret element S_(y(m+1)), but not using the lower-level key generation parameters Q_(yi) for i<l. Similarly, the ciphertext C may be decoded to recover the message Musing the lower-level key generation parameters Q_(zi) for i≧l and the recipient secret element S_(z(n+1)), but not using the lower-level key generation parameters Q_(zi) for i<l.

For instance, in a BasicHIDE scheme (FIGS. 4 and 5), application of dual-HIDE changes the encoding of the message M to generate a ciphertext C=[U₀, U_(l+1), . . . , U_(n+1),V], wherein U_(i)=rP_(zi) for i=0 and for l+1≦i≦n+1, wherein V=M ⊕ H₂(g_(yl) ^(r)), and wherein $g_{yl} = {\frac{\hat{e}\left( {P_{0},S_{y{({m + 1})}}} \right)}{\prod\limits_{i = {l + 1}}^{m + 1}\quad{\hat{e}\left( {Q_{y{({i - 1})}},P_{yi}} \right)}}.}$ The U_(i) factors are calculated in the same way as before, but fewer of them are necessary. However, dual-BasicHIDE does require the sender y to use more key generation parameters Q_(yi) to generate g_(yl) than are necessary to generate g as describe above. This is because the sender's identity is being incorporated into the Encryption algorithm.

The increase in efficiency of the Decryption algorithm is more dramatic. The message M is recovered using $M = {V \oplus {{H_{2}\left( \frac{\hat{e}\left( {U_{0},S_{z{({n + 1})}}} \right)}{\prod\limits_{i = {l + 1}}^{n + 1}\quad{\hat{e}\left( {Q_{z{({i - 1})}},U_{zi}} \right)}} \right)}.}}$

Again, fewer U_(i) parameters are necessary. Similarly, the recipient requires fewer key generation parameters Q_(zi) for dual-HIDE than would otherwise be necessary.

FullHIDE also may be modified to create a dual-FullHIDE scheme. Generation of the ciphertext C in the Encryption algorithm is modified such that C=[U₀, U_(l+1), . . . , U_(n+1), V, W], wherein U_(i)=rP_(zi) for i=0 and for l+1≦i≦n+1. The W and r parameters is still generated the same way, W=E_(H) ₄ _((σ))(M), and the g_(yl) parameter in V=σ ⊕ H₂(g_(yl) ^(r)) is generated using $g_{yl} = {\frac{\hat{e}\left( {P_{0},S_{y{({m + 1})}}} \right)}{\prod\limits_{i = {l + 1}}^{m + 1}\quad{\hat{e}\left( {Q_{y{({i - 1})}},P_{yi}} \right)}}.}$

The Decryption algorithm also is modified in a dual-FullHIDE scheme. The random binary string σ is recovered using $\sigma = {V \oplus {{H_{2}\left( \frac{\hat{e}\left( {U_{0},S_{z{({n + 1})}}} \right)}{\prod\limits_{i = {l + 1}}^{n + 1}\quad{\hat{e}\left( {Q_{z{({i - 1})}},U_{zi}} \right)}} \right)}.}}$ Otherwise, recovery of the message M does not change.

Although these dual-HIDE schemes have been described using PKG_(l) 304 b as the lowest ancestor PKG common to both the sender y and the recipient z, PKG_(l) 304 b may be any common ancestor PKG. The encryption and decryption algorithms are the same. For maximum efficiency however, it is preferable that PKG_(l) 304 b be the lowest common ancestor PKG.

In addition to the increase in efficiency, the dual-HIDE schemes of the present invention also offer increased security by restricting key escrow. In the BasicHIDE and FullHIDE schemes described above, all of the recipient's direct ancestor PKGs are able to decrypt messages to the recipient. However, because the dual-HIDE schemes incorporate the key generation secret of PKG_(l−1) (the immediate parent of PKG_(l)), which is unknown to the common ancestor PKGs above PKG_(l−1), those common ancestor PKGs are not able to decrypt messages between the sender y and the recipient z. The immediate parent of PKG_(l) 304 b is still able to decrypt messages, however, because it knows its own key generation secret.

Key escrow may be further restricted such that even the immediate parent of PKG, may not decrypt messages between the sender y and the recipient z. This may be accomplished by obscuring PKG_(l)'s private key in the process of generating private keys for the sender y and the recipient z (or private keys for children of PKG_(l) that are ancestors of the sender y and the recipient z). For instance, PKG_(l) 304 b may easily change its private key by setting S_(l)′:=S_(l)+bP_(l), and Q_(l−1)′:=Q_(l−1)+bP₀, for some random b ε Z/qZ. The new private key S_(l)′ is just as effective, but is unknown to PKG_(l)'s immediate parent. Accordingly, no PKGs above PKG_(l) are able to decode messages encrypted to the recipient z. More specifically, only ancestors of the recipient z that are within PKG_(l)'s domain are able to decrypt messages to the recipient z.

When PKG_(l) 304 b changes its private key by setting S_(l)′:=S_(l)+bP_(l), and Q_(l−1)′:=Q_(l−1)+bP₀, the new private key is still related to PKG_(l−1)'s key generation secret s_(l−1), because the new private key is derived from a private key generated by PKG_(l−1) using s_(l−1). In general, in all of the schemes discussed herein, a user or PKG may change its own secret element S_(z(n+1)) and key generation parameters Q_(zi) for 1≦i≦n by choosing values for b_(i) for 1≦i≦n and setting $S_{z{({n + 1})}}^{\prime}:={S_{z{({n + 1})}} + {\sum\limits_{i = 1}^{n}\quad{b_{i}P_{z{({i + 1})}}}}}$ and Q_(zi)′:=Q_(zi)+b_(i)P₀ for 1≦i≦n. For purposes of the present invention, however, this new private key is still considered to be related to the original private key, and is thus related to the original values of the key generation secrets s_(zi). Dual-HIDE Scheme with more Efficient Encryption or Decryption

In the dual-HIDE schemes described above, it is possible to decrease by one the number of values of the pairing that the encrypter must compute without increasing the number of values of the pairing that the decrypter must compute. For instance, the dual-BasicHIDE Encryption algorithm described above may be modified such that the ciphertext C=[rP₀,r(P_(y(l+1))−P_(z(l+1))), rP_(z(l+2)), . . . , rP_(z(n+1)),M ⊕ H₂(g_(y(l+1)) ^(r))], where $g_{y{({l + 1})}} = {\frac{\hat{e}\left( {P_{0},S_{y{({n + 1})}}} \right)}{\prod\limits_{i = {l + 2}}^{m}\quad{\hat{e}\left( {Q_{y{({i - 1})}},P_{yi}} \right)}} = {{\hat{e}\left( {P_{0},S_{y{({l + 1})}}} \right)}.}}$ If the ciphertext is represented as C=[U₀,U_(l+1), . . . , U_(n+1),V], then it may be decrypted using $M = {V \oplus {{H_{2}\left( \frac{{\hat{e}\left( {U_{0},S_{z{({n + 1})}}} \right)}{\hat{e}\left( {U_{l + 1},Q_{zl}} \right)}}{\prod\limits_{i = {l + 2}}^{n}\quad{\hat{e}\left( {Q_{z{({i - 1})}},U_{i}} \right)}} \right)}.}}$

Likewise, it is possible to decrease by one the number of values of the pairing that the decrypter must compute without increasing the number of values that the encrypter must compute. For instance, the dual-BasicHIDE Encryption algorithm may be modified such that the ciphertext C=[rP₀,rP_(y(l+2)), . . . , rP_(y(n)),M ⊕ H₂(g_(z(l+1)) ^(r))], where $g_{z{({l + 1})}} = {\frac{{\hat{e}\left( {P_{0},S_{y{({m + 1})}}} \right)}{\hat{e}\left( {Q_{yl},P_{z{({l + 1})}},P_{{y{({l + 1})}})}} \right)}}{\prod\limits_{i = {l + 2}}^{m}\quad{\hat{e}\left( {Q_{y{({i - 1})}},P_{yi}} \right)}} = {{\hat{e}\left( {P_{0},S_{z{({l + 1})}}} \right)}.}}$ If the ciphertext is represented as C=[U₀, U_(l+2), . . . , U_(n),V], then it may be decrypted using $M = {V \oplus {{H_{2}\left( \frac{\hat{e}\left( {U_{0},S_{z{({n + 1})}}} \right)}{\prod\limits_{i = {l + 2}}^{n}\quad{\hat{e}\left( {Q_{z{({i - 1})}},U_{i}} \right)}} \right)}.}}$ Authenticated Lower-Level Root PKGs

The efficiencies of the dual-HIDE schemes described above may be extended to message senders who are outside the hierarchy by creating an authenticated lower-level root PKG. To “authenticate” the lower-level PKG, the root PKG may issue an additional parameter, such as a random message M′. The lower-level PKG then “signs” M′, generating the signature Sig=S_(zl)+s_(zl)P_(M′), where S_(l) is the lower-level PKG's private key, and s_(t) is its lower-level key generation secret. The lower-level PKG also publishes Q_(i) for 1≦i≦t.

Taking advantage of the authenticated lower-level root PKG, a sender outside the hierarchy may send an encrypted message to the recipient z without computing public elements P_(zi) for all n of the recipient's ancestor PKGs. Rather, the sender may use the parameters for the lower-level authenticated root PKG to encrypt the message more efficiently. In particular, the sender computes P_(zi)=H₁(ID₁, . . . , ID_(zi)) ε G₁ for l+1≦i≦n+1. The sender then chooses a random r ε Z/qZ, and generates the ciphertext C=[rP₀,rP_(z(l+1)), . . . , rP_(z(n+1)), M ⊕ H₂(g_(zl) ^(r))], where ${g_{zl} = {\frac{\hat{e}\left( {P_{0},{Sig}} \right)}{\hat{e}\left( {{s_{zl}P_{0}},P_{M^{\prime}}} \right)} = {{\hat{e}\left( {P_{0},S_{zl}} \right)}.}}}\quad$ Letting the received ciphertext C=[U₀, U_(l+1), . . . , U_(n+1), V], the recipient may then decrypt the ciphertext to recover the message ${{M = {V \oplus {H_{2}\left\lbrack \frac{\hat{e}\left( {U_{0},S_{z{({n + 1})}}} \right)}{\prod\limits_{i = {l + 1}}^{n + 1}\quad{\hat{e}\left( {Q_{z{({i - 1})}},U_{i}} \right)}} \right\rbrack}}},}\quad$ where S_(z(n+1)) is the recipient's private key. Distributed PKGs

To further protect the key generation secrets of the HIDE schemes described above, and to make the schemes robust against dishonest PKGs, the key generation secrets and private keys may be distributed using known techniques of threshold cryptography.

More Efficient Encryption

The efficiency of encryption for the HIDE schemes described above may be increased by merging the highest two levels of the hierarchy into a single root PKG. In that case, g=ê(Q₀,P₁) is included in the system parameters. This saves encrypters the task of computing the value of this pairing. However, the decrypters must compute one extra pairing (as a result of being one level lower down the tree).

HIDS Schemes

Turning now to the signature, or HIDS, schemes of the present invention, FIG. 7 shows a flow diagram illustrating a method of generating and verifying a digital signature according to another presently preferred embodiment of the invention. The method is performed in a HIDS system including a plurality of PKGs. The PKGs include at least a root PKG and n lower-level PKGs in the hierarchy between the root PKG and the sender, or signer, wherein n≧1. In block 702, the root PKG selects a root key generation secret known only to the root PKG. The root key generation secret may be used to generate private keys for PKGs or users below the root PKG in the hierarchy. The root PKG then generates a root key generation parameter based on the root key generation secret in block 704. The lower-level PKGs select lower-level key generation secrets in block 706. The lower-level key generation associated with a given lower-level PKG may be used to generate private keys for PKGs or users below the associated lower-level PKG in the hierarchy. Like the root key generation secret, each of the lower-level key generation secrets is known only to its associated lower-level PKG. In block 708, lower-level key generation parameters are generated for each of the n lower-level PKGs. Each of the lower-level key generation parameters is generated using at least the lower-level key generation secret for its associated lower-level PKG.

In block 710, a lower-level PKG generates a private key for the recipient such that the private key is related to at least one of the n lower-level key generation secrets. For instance, the sender's private key may be related at least to the lower-level key generation secret of the PKG that issued the private key to the recipient. Preferably, however, the recipient's private key may be related to all n of its ancestral PKG's lower-level key generation secrets, as well as the root key generation secret. In block 712, the sender uses at least its private key to sign the message and generate the digital signature. The recipient, or verifier, then verifies the digital signature in block 714 using at least one of the lower-level key generation parameters. For instance, the signature may be verified using only the root key generation parameter. Alternatively, one or more of the lower-level key generation parameters also may be used.

FIG. 8 shows a flow diagram illustrating a method of generating and verifying a digital signature Sig of a digital message M communicated between a sender y and a recipient z according to another presently preferred embodiment of the invention. The sender y 306 is m+1 levels below the root PKG in the hierarchy, as shown in FIG. 3, and is associated with the ID-tuple (ID_(y1), . . . , ID_(y(m+1))). The sender's ID-tuple includes identity information ID_(y(m+1)) associated with the sender, as well as identity information ID_(yi) associated with each of its m ancestral lower-level PKGs in the hierarchy. The method begins in block 802 by generating first and second cyclic groups G₁ and G₂ of elements. In block 804, a function ê is selected, such that the function ê is capable of generating an element of the second cyclic group G₂ from two elements of the first cyclic group G₁. The function ê preferably is an admissible pairing, as described above. A root generator P₀ of the first cyclic group G₁ is selected in block 806. In block 808, a random root key generation secret s₀ associated with and known only to the root PKG 302 is selected. Preferably, s₀ is an element of the cyclic group Z/qZ. A root key generation parameter Q₀=s₀P₀ is generated in block 810. Preferably, Q₀ is an element of the first cyclic group G₁. In block 812, a first function H₁ is selected such that H₁ is capable of generating an element of the first cyclic group G₁ from a first string of binary digits. A second function H₃ is selected in block 814, such that H₃ is capable of generating a second string of binary digits from an element of the second cyclic group G₂. The functions of blocks 802 through 814 are part of the HIDS Root Setup algorithm described above, and preferably are performed at about the same time. By way of example, functions such as those disclosed in Boneh-Franklin may be used as H₁ and H₃. In fact, the functions H₁ and H₃ may be exactly the same function. However, there is a potential pitfall. An attacker may try to get the signer to sign M=ID_(t), wherein ID_(t) represents an actual identity. In this case, the signer's signature may actually be a private key, which thereafter may be used to decrypt messages and forge signatures. This pitfall may be avoided, however, by using some expedient—such as a bit prefix or a different function for H₃—that distinguishes between signing and private key extraction.

The next series of blocks (blocks 816 through 824) show the functions performed as part of Lower-level Setup algorithm. In block 816, a public element P_(yi) is generated for each of the sender's m ancestral lower-level PKGs. Each of the public elements, P_(yi)=H₁(ID₁, . . . , ID_(yi)) for 1≦i≦m, preferably is an element of the first cyclic group G₁. Although represented in a single block, generation of all the public elements P_(yi) may take place over time, rather than all at once.

A lower-level key generation secret s_(yi) is selected (block 818) for each of the sender's m ancestral lower-level PKGs 304 a,b,d. The lower-level key generation secrets s_(yi) preferably are elements of the cyclic group Z/qZ for 1≦i≦m, and each lower-level key generation secret s_(yi) preferably is known only to its associated lower-level PKG. Again, although represented in a single block, selection of all the secrets s_(yi) may take place over time, rather than all at once.

A lower-level secret element S_(yi) is generated (block 820) for each of the sender's m ancestral lower-level PKGs. Each lower-level secret element, S_(yi)=S_(y(i−1))+S_(y(i−1))P_(yi) for 1≦i≦m, preferably is an element of the first cyclic group G₁. Although represented in a single block like the public elements P_(yi) and the secrets S_(yi), generation of all the secret elements S_(yi) may take place over time, rather than all at once. For purposes of these iterative key generation processes, S₀ preferably is defined to be the identity element of G₁.

A lower-level key generation parameter Q_(yi) also is generated (block 824) for each of the sender's m ancestral lower-level PKGs. Each of the key generation parameters, Q_(yi)=s_(yi)P₀ for 1≦i≦m, preferably is an element of the first cyclic group G₁. Again, although represented in a single block, generation of all the key generation parameters Q_(yi) may take place over time, rather than all at once.

The functions of the next two blocks (blocks 824 and 826) are performed as part of the Extraction algorithm described above. A sender public element P_(y(m+1)) associated with the sender y is generated in block 824. The sender public element, P_(y(m+1))=H₁(ID_(y1), . . . , ID_(y(m+1))), preferably is an element of the first cyclic group G₁. A sender secret element S_(y(m+1)) associated with the sender y is then generated in block 826. The sender secret element ${S_{y{({m + 1})}} = {{S_{ym} + {s_{ym}P_{y{({m + 1})}}}} = {\sum\limits_{i = 1}^{m + 1}\quad{s_{y{({i - 1})}}P_{yi}}}}},$ also preferably is an element of the first cyclic group G₁.

For convenience, the first function H₁ optionally may be chosen to be an iterated function so that, for example, the public points P_(i) may be computed as H₁(P_(y(i−1)), ID_(yi)) rather than H₁(ID₁, . . . , ID_(yi)).

The last two blocks shown in FIG. 8 (blocks 828 and 830) represent the Signing and Verification algorithms described above. In block 828, the message M is signed to generate a digital signature Sig. The signing preferably uses at least the sender secret element S_(y(m+1)). The digital signature Sig is then verified in block 830. The verification preferably uses at least the root key generation parameter Q₀ and the lower-level key generation parameters Q_(yi). The specific use of these parameters and elements in the signing of the message M and verification of the digital signature Sig will now be discussed with reference to FIG. 9.

FIG. 9 shows a flow diagram illustrating a method of generating and verifying a digital signature Sig of a digital message M communicated between a sender y and a recipient z according to another presently preferred embodiment of the invention. In this scheme the Root Setup, Lower-level Setup, and Extraction algorithms are the same as for the embodiment shown in blocks 802 through 826 of FIG. 8. Accordingly, the flow diagram of FIG. 9 begins with the selection of a sender key generation secret s_(y(m+1)), known only to the sender y, in block 927 a. A sender key generation parameter Q_(y(m+1)) associated with the sender is generated in block 927 b using the formula Q_(y(m+1))=S_(y(m+1))P₀. The Signing algorithm then begins with the sender generating a message element P_(M)=H₃(ID_(y1), . . . , ID_(y(m+1)), M) in block 928 a. The message element P_(M) preferably is a member of the first cyclic group G₁. The digital signature Sig itself is generated in block 928 b using the formula Sig=S_(y(m+1))+s_(y(m+1))P_(M). The recipient verifies the digital signature Sig (block 930) by confirming that the formula $\frac{\hat{e}\left( {P_{0},{Sig}} \right)}{{\hat{e}\left( {Q_{y{({m + 1})}},P_{M}} \right)}{\prod\limits_{i = 2}^{m + 1}\quad{\hat{e}\left( {Q_{y{({i - 1})}},P_{yi}} \right)}}} = {\hat{e}\left( {Q_{0},P_{1}} \right)}$ is satisfied.

The invention has been described in detail with particular reference to preferred embodiments thereof and illustrative examples, but it will be understood that variations and modifications can be effected within the spirit and scope of the invention. 

1. A computer-implemented encryption method for encrypting a message M for decryption with secret information of an intended recipient entity E which is to use secret information obtained using a value or values generated by one or more of E's parents in a hierarchical cryptographic scheme, the method comprising: (1) generating variable data including a variable value r; (2) forming a first encryption of the message M, wherein the first encryption is decryptable without using E's secret information but using a value g^(r), where g=ê(p ₀ ,p ₁), ê:G₁×G₁→G₂ is a non-degenerate bilinear mapping, where G₁ and G₂ are groups, where ê, G₁, G₂ are pre-defined to be independent of M; p₀ is pre-defined to be independent of M and of E's ID, and P₁ is independent of E's ID but depends on ID's of one or more of E's parents; (3) forming a first ciphertext portion dependent on E's ID, wherein the first ciphertext portion is formed as a function of one or more inputs including the variable data and E's ID but not including E's secret information, but the value g^(r) is computable as a function of one or more inputs including the first ciphertext portion and E's secret information; (4) outputting a ciphertext formed using the first ciphertext portion and the first encryption of the message M.
 2. The method of claim 1 wherein the first ciphertext portion does not depend on M.
 3. The method of claim 1 wherein the hierarchical encryption scheme is such that: a root entity E₀ generates its secret integer so; each entity E_(i) which is E's parent at a hierarchical level i>0, generates a respective secret integer s_(i); for each integer iε[1,t], where t>1 is a hierarchical level of E and where E_(t)=E, each entity E_(i) obtains a value equal to S_(i−1)+s_(i−1)P_(i), and sets E_(i)'s secret S_(i) to said value equal to S_(i−1)+s_(i−1)P_(i) or to a value obtained by modifying said value equal to S_(i−1)+s_(i−1)P_(i), where each P_(i) is a public function of one or more ID's including E_(i)'s ID, and S₀ is the identity element of G₁; each entity E_(i)(iε[1,t−1]) generates an element Q_(i) as Q_(i)=s_(i)P₀, each entity E_(i)(iε[2,t−1]) obtains a set of elements Q_(k) for all kε[1,i−1] and either modifies one or more of Q_(k) or leaves them unmodified to correspond to S_(i), and each entity E_(i)(iε[1,t−1]) provides all of the elements Q_(k) (kε[1,i]) to E_(i+1).
 4. The method of claim 3 wherein for at least one value iε[2,t−1], E_(i) sets S_(i) to equal to S_(i−1)+s_(i−1)P_(i) and the elements Q_(k) are unmodified by E_(i).
 5. The method of claim 3 wherein for at least one value iε[2,t−1]: E_(i) generates one or more variable integer values b_(c) for one or more integer values cε[1,i−1] and generates S_(i) as a sum of the values b_(c)p_(c+1) with said value equal to S_(i−1)+s_(i−1)P_(i); and E_(i) modifies the elements Q_(k) by replacing, for each said c, the element Q_(c) with Q_(c)+b_(c)P₀.
 6. The method of claim 3 wherein the first ciphertext portion comprises a plurality of values of a form rF where each F is a linear expression in one or more of the elements P₀, . . . , P_(t) such that the values F together define all of the values P₀, . . . , P_(t) except P_(j); g=ê(Q ₀ ,P _(j)) for some jε[1,t−1], and g^(r) is computable as ${g^{r} = \frac{\hat{e}\left( {U_{0},S_{t}} \right)}{\prod\limits_{{1 \leq i \leq t},{i \neq j}}^{\quad}\quad{\hat{e}\left( {Q_{i - 1},U_{i}} \right)}}}\quad$ where U₀=rP₀, each U_(i)=rP_(i).
 7. The method of claim 6 wherein the values F are the values P_(i) for all iε[0,t] such that i≠1, and j=1.
 8. The method of claim 3 wherein E computes g as: ${g = \frac{\hat{e}\left( {P_{0},{Sig}} \right)}{\hat{e}\left( {{s_{l}P_{0}},P_{M^{\prime}}} \right)}}\quad$ where Sig=S_(l)+s_(l)P_(M′) is provided to E by another entity, l being a selected integer, l<t, where S_(l) is unavailable to E, and P_(M′) is available to E, and the product s_(l)P₀ is also available to E as Q_(l); where g^(r) is computable as ${g^{r} = \frac{\hat{e}\left( {U_{0},S_{t}} \right)}{\prod\limits_{i = {l + 1}}^{t}\quad{\hat{e}\left( {Q_{i - 1},U_{i}} \right)}}}\quad$ where U_(i)=rP_(i) for each iε[l+1,t]; and the first ciphertext portion comprises a plurality of values of a form rF such that the values F together define all P₀ and all of P_(l+1), . . . P_(t).
 9. The method of claim 8 where g=ê(P₀,S_(l)).
 10. The method of claim 3 wherein: the method is performed on behalf of an entity E_(ym) which is in the hierarchical encryption scheme at a level m and which is a child of E_(l) for some l<t; each entity E_(yi) which is E_(ym)'s parent at a hierarchical level i>0, generates a respective secret integer s_(yi); for each integer iε[1,m], each entity E_(yi) obtains E_(yi)'s secret S_(yi) from a value equal to S_(y(i−1))+s_(y(i−1))P_(yi), where each P_(yi) is a public function of one or more ID's including E_(yi)'s ID, and S_(y0) is the identity element of G₁; each entity E_(yi)(iε[1,m−1]) generates an element Q_(yi) as Q_(yi)=s_(yi)P₀, obtains a set of elements Q_(yk) for all kε[1,i−1], and provides either modified or unmodified elements Q_(yk) and Q_(yi) to E_(y(i+1)).
 11. The method of claim 10 wherein: ${g = \frac{\hat{e}\left( {P_{0},S_{ym}} \right)}{\prod\limits_{i = {l + 1}}^{m}\quad{\hat{e}\left( {Q_{y{({i - 1})}},P_{yi}} \right)}}}\quad$ and g^(r) is computable as ${g^{r} = \frac{\hat{e}\left( {U_{0},S_{t}} \right)}{\prod\limits_{i = {l + 1}}^{t}\quad{\hat{e}\left( {Q_{i - 1},U_{i}} \right)}}}\quad$ where U_(i)=rP_(i) for each iε[l+1,t]; and the first ciphertext portion comprises a plurality of values of a form rF such that the values F together define all P₀ and all of P_(l+1), . . . , P_(t).
 12. The method of claim 11 wherein s_(yl) is generated as a variable value separately from s_(i).
 13. The method of claim 10 wherein s_(yl)=s_(l), and: ${g = \frac{\hat{e}\left( {P_{0},S_{ym}} \right)}{\prod\limits_{i = {l + 2}}^{m}\quad{\hat{e}\left( {Q_{y{({i - 1})}}^{\prime},P_{yi}} \right)}}}\quad$ and g^(r) is computable as $g^{r} = \frac{\hat{e}\left( {U_{0},{S_{t}{\hat{e}\left( {U_{l + 1},Q_{l}^{\prime}} \right)}}} \right)}{\prod\limits_{i = {l + 2}}^{t}\quad{\hat{e}\left( {Q_{i - 1}^{\prime},U_{i}} \right)}}$ where U_(l+1)=r(P_(y(l+1))−P_(l+1)), and U_(i)=rP_(i) for i=0 and each iε[l+2,t]; and the first ciphertext portion comprises a plurality of values of a form rF such that the values F together define P₀, P_(y(l+1))−P_((l+1)), and all of P_(l+2), . . . , P_(t).
 14. The method of claim 3 wherein for each i>1, P_(i) is a function of (ID₁, . . . , ID_(i)).
 15. A computer-implemented encryption method for encrypting a message M for decryption by an intended recipient entity E which is to use secret information obtained using a value or values generated by one or more of E's parents in a hierarchical cryptographic scheme, the method comprising: generating a variable value; combining the variable value with E's ID to obtain a first ciphertext portion which is independent of the message M but is dependent on E's ID; generating a value g^(r) equal to a function of the variable value and a value of a bilinear non-degenerate mapping ê:G₁×G₁→G₂ on two elements one of which is a function of ID's of one or more of E's parents, wherein G₁, G₂ are groups, the value g^(r) being independent of E's ID; generating a second ciphertext portion depending on the message M and the value g_(r); generating a ciphertext which includes the first and second ciphertext portions.
 16. The method of claim 15 wherein the second ciphertext portion is independent of E's ID but dependent on said ID's of the one or more of E's parents.
 17. The method of claim 15 wherein E's secret information comprises a sum S of (ii) secret information of one or more of E's parents and (ii) one or more terms; and the value g^(r) is a function of a ratio A/B, where: A is a product of one or more values at least one of which is a value of the bilinear mapping on two elements one of which is S and the other one of which is a function of the first ciphertext portion; and B is a product of one or more values each of which is a value of the bilinear mapping on two elements one of which is a function of the first ciphertext portion and the other one of which is a function of secret information used to define the one or more terms.
 18. The method of claim 17 wherein A and B each depend on the one or more terms but A/B does not depend on the one or more terms and does not depend on S but depends on the secret information of the one or more of the E's parents whose ID's were used to obtain the value g^(r).
 19. The method of claim 15 wherein the one or more of the E's parents whose ID's were used to obtain the value g^(r) is a single parent.
 20. The method of claim 15 wherein one or more of the E's parents whose ID's were used to obtain the value g^(r) are common parents of E and an entity on whose behalf the encryption is being performed.
 21. The method of claim 20 wherein E's secret information comprises a sum S of (i) secret information associated with the common parent and (ii) one or more terms; and the value g^(r) is a function of a ratio A/B, where: A is a product of one or more values at least one of which is a value of the bilinear mapping on two elements one of which is S and the other one of which is a function of the first ciphertext portion; and B is a product of one or more values each of which is a value of the bilinear mapping on two elements one of which is a function of the first ciphertext portion and the other one of which is a function of secret information used to define the one or more terms.
 22. A computer-implemented encryption method comprising: obtaining a specification of elements Q₀,P₀,P₁, . . . , P_(t) of a group G₁ and of a non-degenerate bilinear mapping ê:G₁×G₁→G₂ where G₂ is a group and t>1; obtaining a variable integer value r; obtaining a value g^(r), where g=ê(Q₀,P_(j)) for a selected integer jε[1,t]; obtaining a message M to be encrypted; obtaining a value W₁ from g^(r) and M such that W₁ incorporates a first encryption of M which is decryptable using g^(r); obtaining values U_(i)=rP_(i) for at least each iε[0,t] such that i≠j; and providing a ciphertext C which is an encryption of M, the ciphertext C comprising said values U_(i) and the value W₁.
 23. The method of claim 22 where j<t; where for each integer iε[1,t], P_(i) is a function of ID_(i) which is an ID of a respective entity E_(i), but P_(i) is independent of any ID_(k) for any k>i; wherein the method is part of a hierarchical encryption scheme according to which: (i) each E_(i) (i>0) generates a respective integer s_(i) and obtains E_(i)'s secret S_(i) from a value equal to S_(i−1)+s_(i−1)P_(i), where S₀ is the identity element of G₁ and s₀ is generated by a root entity E₀; and $\begin{matrix} {g^{r} = \frac{\hat{e}\left( {U_{0},S_{t}} \right)}{\prod\limits_{{1 \leq i \leq t},{i \neq j}}^{\quad}\quad{\hat{e}\left( {Q_{i - 1},U_{i}} \right)}}} & ({ii}) \end{matrix}$ where {Q_(i−1)|1≦i≦t, i≠j} is a set of elements of G₁ and is available to E_(t).
 24. The method of claim 23 where Q₀=s₀P₀.
 25. The method of claim 23 further comprising sending the ciphertext C to E_(t) for decryption by E_(t), the decryption is operable to be performed by E_(t) using S_(t) but not S_(j).
 26. The method of claim 25 wherein the values S_(t) and {Q_(i−1)|1≦i≦t, i≠j} are such that: ${S_{t} = \sum\limits_{i = 1}^{t}}\quad$ s_(i−1)P_(i) for some integers s_(i−1); and Q_(i−1)=s_(i−1)P₀ for all i>1.
 27. The method of claim 25 wherein for each i>1, P_(i) is a function of (ID₁, . . . , ID_(i)).
 28. The method of claim 23 where j=1.
 29. The method of claim 23 where W₁ is either (i) a predefined function of g^(r) and M, or (ii) W₁=[V,W] where V is a predefined function of g^(r) and a value a such that or is recoverable from V given g^(r), and W is an encryption of M under a key equal to a function of σ.
 30. The method of claim 23 where in the hierarchical encryption scheme: the entity E₀ generates E₀'s secret integer value s₀ and also generates P₀, sets Q₀=s₀P₀, and makes P₀ and Q₀ available for encryption; each E_(i) (1≦i≦t−1) generates Q_(i) as Q_(i)=s_(i)P₀; and each E_(i) (1≦i≦t) obtains at least all the elements Q₀, . . . , Q_(i−1) other than Q_(j−1).
 31. The method of claim 30 wherein each E_(i)(2≦i≦t) obtains at least all the elements Q₁, . . . , Q_(i−1) other than Q_(j−1) from E_(i−1), and each E_(i) (1≦i≦t−1) either provides at least all the elements Q₁, . . . Q_(i) other than Q_(j−1) to E_(i+1) unchanged or increments Q_(i−1) by bP₀ and generates S_(i) by incrementing said value equal to S_(i−1)+s_(i−1)P_(i) by bP_(i), where b is a variable integer value generated by E_(i).
 32. The method of claim 30 wherein to decrypt the ciphertext C, E_(t) uses S_(t) to recover g^(r) as $g^{r} = \frac{\hat{e}\left( {U_{0},S_{t}} \right)}{\prod\limits_{{1 \leq i \leq t},{i \neq j}}^{\quad}\quad{\hat{e}\left( {Q_{i - 1},U_{i}} \right)}}$
 33. The method of claim 23 wherein each E_(i) (i>0) is an immediate child of E_(i−1), and for at least one i>0, E_(i−1) has one or more other immediate children but provides the same value S_(i−1)+s_(i−1)P_(i) to each of its immediate children in operation (i) performed for each of the immediate children.
 34. The method of claim 23 wherein each E_(i) (i>0) is an immediate child of E_(i−1), and for at least one i>0, E_(i−1) has one or more other immediate children and separately generates a variable value s_(i−1) for each of the immediate children for use in the respective value S_(i−1)+s_(i−1)P_(i).
 35. A computer-implemented decryption method for recovering a message M from a ciphertext C on behalf of an entity E possessing E's secret information obtained using a value or values obtained from one or more of E's parents in a hierarchical cryptographic scheme, E's secret information comprising an element S(E) of a group G₁, wherein S(E) is a linear combination of first members of the group G₁, each first member being associated with a respective integer coefficient in said linear combination, the first members comprising one or more first elements of the group G₁ each of which depends on E's ID, the first members comprising one or more second elements of the group G₁ each of which is independent of E's ID, the method comprising: (1) computing a value g_(r) from: the ciphertext C; the element S(E); and a set of one or more elements Q of the group G₁; wherein each element Q is independent of the ciphertext C but is dependent on one or more of the first and second elements' coefficients, wherein the value g_(r) does not depend on E's ID but depends on one or more of the second elements' coefficients, the value g^(r) being computed as a function of a ratio A/B, wherein: (i) A is a first product of one or more elements of a group G₂, the one or more elements of the first product comprising an element which is a value of a bilinear non-degenerate mapping ê:G₁×G₁→G₂ on two elements at least one of which depends on S(E) and at least one of which depends on the ciphertext C; and (ii) B is a second product of one or more elements of the group G₂, the one or more elements of the second product including one or more elements each of which is a value of the mapping ê on two elements at least one of which depends on one or more of the elements Q and at least one of which depends on the ciphertext C; (2) using the value g^(r) to recover the message M from C.
 36. The method of claim 35 wherein operation (2) does not use any information on the first elements' coefficients other than information present in A and/or B and/or the ciphertext C.
 37. The method of claim 35 wherein operation (2) does not use any information on E's ID other than information present in A and/or B and/or the ciphertext C.
 38. The method of claim 37 wherein the value g^(r) depends on ID's of one or more of E's parents.
 39. The method of claim 35 wherein the first elements' coefficients are each a secret of E's parent and is unavailable to E for decryption.
 40. The method of claim 39 wherein the second elements' coefficients are each a secret of E's parent and is unavailable to E for decryption.
 41. The method of claim 35 wherein one or more of the second elements depend on ID's of one or more of E's parents.
 42. The method of claim 35 wherein ${{S(E)} = {\sum\limits_{i = 1}^{t}\quad{s_{i - 1}P_{i}}}}\quad$ where: t>1 is E's hierarchical level; {P_(i)} are said first members and {s_(i−1)} are said coefficients; E_(t)=E and E_(i) is E's parent at hierarchical level i for each iε[1,t−1], and where for each i<t, P_(i) is a function of ID_(i) which is an ID of a respective entity E_(i), but P_(i) is independent of any ID_(k) for any k>i; and the one or more elements Q are one or more elements Q_(i)=s_(i−1)P₀ for one or more integers iε[1,t], where P₀ is a predefined element of the group G₁.
 43. The method of claim 42 wherein ${g_{r} = \frac{\hat{e}\left( {U_{0},{S(E)}} \right)}{\prod\limits_{{1 \leq i \leq t},{i \neq j}}^{\quad}\quad{\hat{e}\left( {Q_{i - 1},U_{i}} \right)}}}\quad$ where: j is a selected integer in [1,t−1]; the elements U₀, U_(i)(1≦i≦t}, i≠j) are obtained from the ciphertext C.
 44. The method of claim 43 wherein each U_(i)=rP_(i), where r is an integer not available to E.
 45. The method of claim 42 wherein one of the following conditions (a) or (b) is true: $\begin{matrix} {{g_{r} = \frac{\hat{e}\left( {U_{0},{S(E)}} \right)}{\prod\limits_{i = {l + 1}}^{t}\quad{\hat{e}\left( {Q_{i - 1},U_{i}} \right)}}}\quad} & (a) \end{matrix}$ where: l is a selected integer in [2,t−1]; the elements U₀, U_(i)(i=l+1, . . . , t) are obtained from the ciphertext C. $\begin{matrix} {{g_{r} = \frac{{\hat{e}\left( {U_{0},S_{t}} \right)}{\hat{e}\left( {U_{l + 1},Q_{l}} \right)}}{\prod\limits_{i = {l + 2}}^{t}\quad{\hat{e}\left( {Q_{i - 1},U_{i}} \right)}}}\quad} & (b) \end{matrix}$ where: l is a selected integer in {2, t−2}; the elements U₀, U_(i)(i=l+2, . . . , t) are obtained from the ciphertext C.
 46. The method of claim 42 wherein the hierarchical encryption scheme is such that: each entity E_(i)(0≦i<t) generates a respective secret integer s_(i); for each integer iε[1,t], each entity E_(i) obtains E_(i)'s secret s_(i) from a value equal to S_(i−1)+s_(i−1)P_(i), where S(E)=S_(t) and where S₀ is the identity element of G₁; each entity E_(i) (iε[1,t−1]) generates an element Q_(i) as Q_(i)=s_(i)P₀, each entity E_(i)(iε[2,t−1]) obtains a set of elements Q_(k) for all kε[1,i−1] and either modifies one or more of Q_(k) or leaves them unmodified to correspond to S_(i), and each entity E_(i)(iε[1,t−1]) provides all of the elements Q_(k)(kε[1,i]) to E_(i+1).
 47. A computer-implemented decryption method for a hierarchical decryption system, the method comprising an entity E_(t) at a level t>1 performing operations of: obtaining a secret S_(t) from a value obtained from an entity E_(t−1) at level t−1; obtaining a specification of a set of elements Q_(i) of a group G₁ for at least each integer iε[0,t] such that t>1 and i is not equal to a selected integer jε[1,t−1], and of a non-degenerate bilinear mapping ê:G₁×G₁→G₂ where G₂ is a group and t>1; obtaining a ciphertext C comprising an element U_(i) of the group G₁ for each iε[0,f] such that i≠j, the ciphertext C also comprising a value W₁; using each said element U_(i) and the secret S_(t) to compute a value g_(r) defined by ${\frac{\hat{e}\left( {U_{0},S_{t}} \right)}{\prod\limits_{{1 \leq i \leq t},{i \neq j}}^{\quad}\quad{\hat{e}\left( {Q_{i - 1},U_{i}} \right)}};}\quad$ combining the value g^(r) with the value W₁ to recover a message encrypted in the ciphertext C.
 48. The method of claim 47 where j<t; wherein the method is part of a hierarchical encryption scheme according to which, for a set of entities E_(i) at respective levels i=0, . . . , t: each E_(i) (i>0) generates a respective integer si and obtains E_(j)'s secret S_(i) from a value equal to S_(i−1)+s_(i−1)P_(i), where S₀ is the identity element of G₁ and s₀ is generated by a root entity E₀, and P_(i) is an element of the group G₁ and is associated with E_(i); each U_(i)=rP_(i), where r is a variable integer generated in encrypting the message; the value W₁ is obtained by combining the message with a value g^(r) where g=ê(Q₀,P_(j)).
 49. The method of claim 48 wherein each P_(i)(i>0) is a function of ID_(i) which is an ID of the entity E_(i), but P_(i) is independent of any ID_(k) for any k>i;
 50. The method of claim 49 wherein for each i>1, P_(i) is a function of (ID₁, . . . , ID_(i)).
 51. The method of claim 48 where Q₀=s₀P₀.
 52. The method of claim 48 wherein the message is recovered without using S_(j).
 53. The method of claim 48 wherein the values S_(t) and {Q_(i−1)|1≦i≦t, i≠j} are such that: ${S_{t} = {\sum\limits_{i = 1}^{t}\quad{s_{i - 1}^{\prime}P_{i}}}}\quad$ for some integers s_(i−1)′; and Q_(i−1)=s′_(i−1)P₀ for all i>1.
 54. The method of claim 48 where j=1.
 55. The method of claim 48 where W₁ is either (i) a predefined function of g^(r) and M, or (ii) W₁=[V,W] where V is a predefined function of g^(r) and a value σ such that σ is recoverable from V given g^(r), and W is an encryption of M under a key equal to a function of σ.
 56. The method of claim 48 where in the hierarchical encryption scheme: the entity E₀ generates E₀'s secret integer value s₀ and also generates P₀, sets Q₀=s₀P₀, and makes P₀ and Q₀ available for encryption and decryption; each E_(i)(1≦i≦t−1) generates Q_(i) as Q_(i)=s_(i)P₀; and each E_(i)(1≦i≦t) obtains at least all the elements Q₀, . . . , Q_(i−1) other than Q_(j−1).
 57. The method of claim 56 wherein each E_(i) (2≦i≦t) obtains at least all the elements Q₁, . . . , Q_(i−1) other than Q_(j−1) from E_(i−1), and each E_(i) (1≦i≦t−1) either provides at least all the elements Q₁, . . . , Q_(i) other than Q_(j−1) to E_(i+1) unmodified or increments Q_(i−1) by bP₀ and generates S_(i) by incrementing said value equal to S_(i−1)+s_(i−1)P_(i) by b_(i)P_(i), where b is a variable integer value generated by E_(i).
 58. A computer-implemented key extraction method for a hierarchical cryptographic scheme comprising a plurality of entities each of which is to be associated with a cryptographic secret value S (“S-value”) which is an element of a group G₁, the method comprising, on behalf of an entity E_(t−1) which is one of said entities and which has a hierarchical level t−1, where t>1 and where a root entity E₀ has a hierarchical level 0: (1) obtaining a first secret value corresponding to E_(t−1) from data received from E_(t−1)'s immediate parent; (2) obtaining E_(t−1)'s S-value S_(t−1) from the first secret value corresponding to E_(t−1); (3) for each of E_(t−1)'s one or more immediate children E_(t): obtaining a value P_(t) which is a function of E_(t)'s ID and is an element of the group G₁; generating data incorporating a value S_(t−1)+s_(t−1)P_(t) as a first secret value corresponding to E_(t), where s_(t−1) is a secret value generated by E_(t−1); and providing the data incorporating the first secret value S_(t−1)+s_(t−1)P_(t) to E_(t) to enable E_(t) to generate E_(t)'s secret S-value.
 59. The method of claim 58 wherein t>2, wherein operation (2) comprises setting S_(t−1) to equal to the first secret value corresponding to E_(t−1), and the method further comprises, on behalf of E_(t−1): for each level i such that 1≦i≦t−2, obtaining an element Q_(i) of the group G₁, each element Q_(i) being associated with a parent of E_(t−1) at level i; for each immediate child E_(t) of E_(t−1), providing the elements Q_(i) (1≦i≦t−2) and Q_(t−1) to E_(t), where Q_(t−1) is generated by E_(t−1) as Q_(t−1)=s_(t−1)P₀, where P₀ is a predefined public element of the group G₁, where P₀ is independent of the child E_(t) and of the entity E_(t−1).
 60. The method of claim 58 wherein t>2, wherein operation (2) comprises generating one or more variable integer values b_(c) for one or more selected values cε=[1,t−2], and setting S_(t−1) to equal to a sum of the values b_(c)P_(c+1) and the first secret value corresponding to E_(t−1); wherein the method further comprises, on behalf of E_(t−1): for each level i such that 1≦i≦t−2, obtaining an element Q_(i) of the group G₁, each element Q_(i) being associated with a parent of E_(t−1) at level i; for each said value c, replacing Q_(c) with Q_(c)+b_(c)P₀; and then for each immediate child E_(t) of E_(t−1): for each immediate child E_(t) of E_(t−1), providing the elements Q_(i) (1≦i≦t−2) and Q_(t−1) to E_(t), where Q_(t−1) is generated by E_(t−1), as Q_(t−1)=s_(t−1)P₀, where P₀ is a predefined public element of the group G₁, where P₀ is independent of the child E_(t) and of the entity E_(t−1).
 61. The method of claim 58 wherein E_(t−1) has more than one immediate children E_(t), and s_(t−1) is generated as a single value for all the children E_(t).
 62. The method of claim 58 wherein E_(t−1) has more than one immediate children E_(t), and a separate value s_(t−1) is generated for each child E_(t).
 63. An apparatus for performing the method of claim
 1. 64. An apparatus for performing the method of claim
 15. 65. An apparatus for performing the method of claim
 22. 66. An apparatus for performing the method of claim
 35. 67. An apparatus for performing the method of claim
 47. 68. An apparatus for performing the method of claim
 58. 